Transfer functions of inverting integrators depend on the values of various resistors and capacitors in the circuit. Because the resistance values of resistors and the capacitive values of capacitors vary independently with temperature, the transfer functions of integrators circuits often vary undesirably with temperature.
FIG. 1A shows an amplifier circuit 100 including an operational amplifier (op-amp) 101, three input resistors 111-113, and a feedback capacitor 121. As configured in FIG. 1A, the filter circuit 100 functions as an inverting voltage integrator circuit. The input resistors 111, 112, and 113 have resistance values of RA, RB, and RC, respectively, and the capacitor 121 has a capacitance value of CA. The inverting input of op-amp 101 is coupled to receive three input voltages V1, V2, and V3 via respective resistors 111, 112, and 113. The non-inverting input of op-amp 101 is coupled to ground potential. The output of op-amp 101, which generates an output voltage V_out, is coupled to the inverting input of op-amp 101 by capacitor 121.
The output voltage of circuit 100 is expressed as:
                              V_out          ⁢                      (            t            )                          =                                            -                              1                                  C                  A                                                      ⁢                                          ∫                0                t                            ⁢                                                (                                                                                                              V                          1                                                ⁡                                                  (                          t                          )                                                                                            R                        A                                                              +                                                                                            V                          2                                                ⁡                                                  (                          t                          )                                                                                            R                        B                                                              +                                                                                            V                          3                                                ⁡                                                  (                          t                          )                                                                                            R                        C                                                                              )                                ⁢                                                                  ⁢                                  ⅆ                  t                                                              +                      V_out            ⁢                          (              0              )                                                          (        1        )            Because V_out is a function of 1/(RA*CA), 1/(RB*CA), and 1/(RC*CA), it is desirable to keep the values of 1/(RA*CA), 1/(RB*CA), and 1/(RC*CA) relatively constant across the range of operating temperatures. However, because capacitors and resistors are typically made of different materials and operate on different principles, capacitance and resistance values vary independently with temperature, and resistor values typically vary much more with temperature than capacitance values. As a result, the overall transfer function of circuit 100 varies with the temperature, which undesirably leads to performance degradation and even system instability under different operating conditions.
The variation of the transfer function with temperature is even more pronounced in certain applications of circuit 100. For example, in audio applications, the resistance values of the input resistors are on the order of mega-Ohms, and thus a high resistivity polysilicon material is typically used instead of normal polysilicon material. This high resistivity polysilicon material has a much larger temperature coefficient than normal polysilicon material, and therefore resistors made of such material vary much more with temperature than resistors made of normal polysilicon material. For example, the resistance values of resistors made of high resistivity polysilicon can vary as much as ±30% within operational temperature limits.
There is a need to control the variation of the transfer function of filter circuits such as integrators across operating temperatures. Although possible to adjust for the temperature variation of the transfer function by adjusting the input resistors 111-113, adjustments must be made to all the input resistors 111-113 at the same time because the ratio between the input resistor values must remain the same in order to maintain the same transfer function. Having multiple variable resistors within the same filter circuit is undesirable because variable resistors are complex and consume large amounts of die area.
It is also possible to compensate for the temperature variation of the transfer function by adjusting the capacitance value of the feedback capacitor 121. However, it is difficult to tune a capacitor to compensate for variations of resistance values. For example, complex calculations must be performed to translate variations in resistance values to a corresponding adjustment in capacitance value that compensates for the variations in the resistance values. Furthermore, because the feedback capacitor 121 is often adjusted to compensate for process variations inherent in the fabrication in integrated circuit (IC) devices, performing additional temperature compensation with the capacitor adds even more complexity to the control of the variable capacitor since two separate compensations must be performed concurrently on the same circuit element.
FIG. 1B shows an amplifier circuit 150 including an op-amp 151, input resistors 152(1)-152(n) having corresponding resistances of r1-rn, and a feedback element 153 having an impedance of Zf. The amplifier circuit 150 can be implemented as an inverting gain stage or an inverting voltage integrator. The amplifier circuit 150 has inputs to receive n input voltages V1-Vn via input resistors 152(1)-152(n) coupled to an inverting input of the op-amp 151. The op-amp 151 has an output to generate an output voltage signal V_out, and has a non-inverting input coupled to ground potential. The feedback element 153 is coupled between the inverting input of the op-amp 151 and the output of the op-amp 151.
The output voltage V_out of the amplifier circuit 150 under a default temperature is given as:
                              V_out          ⁢                      (            s            )                          =                                            -                              Z                f                                      ⁢                                          ∑                                  k                  =                  1                                n                            ⁢                                                                    V                    k                                    ⁡                                      (                    s                    )                                                                    r                                                            k                      ⁢                      _                                        ⁢                    0                                                                                =                                    -                              Z                f                                      ⁢                                          ∑                                  k                  =                  1                                n                            ⁢                                                g                                                            k                      ⁢                      _                                        ⁢                    0                                                  ⁢                                                      V                    k                                    ⁡                                      (                    s                    )                                                                                                          (        2        )            where gk—0=1/rk—0, and rk—0 represents the resistance of a corresponding resistor under the default temperature. For example, resistor 152(2) has a resistance of r2—0 under the default temperature.
Similar to circuit 100 of FIG. 1A, the output of the amplifier circuit 150 is subject to variations due to temperature. Furthermore, it is difficult to adjust for the temperature variations by individually tuning resistors 152(1)-152(n).